Re: What is your philosophy? Tracking or Static?
On May 25, 2011, at 10:06 AM, Dominique Lohez wrote:
>
>
> George,
> IMHO your examples are not illustrative of argument in favor of some philosophy.
> They show how problems must be reformulated when interval arithmetic is aimed
> Corliss, George a écrit :
>> Dan,
>>
>> Thank you for the useful characterization. I, too, began assuming tracking, but I am coming more to the static view.
>>
>> Early in AD, a frequently cited difficulty was scaling:
>>
>> x is a vector
>> s = max(|x|)
>> x = x / s // Normalize
>> y = lots of computations
>> y = y * s // Rescale back
>>
>> Is y a differentiable function of x? AD is forced to assume not. Intervals can help decide this question more precisely, but the analogy is whether we care about the history or only the result.
>>
> When x is replaced with a box X, the scaling factor must be calculated for the box as a whole, so it a constant for all the vector in the box.
> So the problematic dependence of s with a particular vector is removed.
I ABSOLUTELY agree this example is addresses better by intervals than by AD. My point is that we are not alone in computing results that are disappointingly pessimistic. Disappointing in the sense that a mathematician knows better than the software.
>
>> Or, suppose
>> x is a scalar
>> f(x) = sign(x)
>> y = 0 * f(x)
>>
>> Is y a continuous function of x?
>>
> This problem is ill-posed.
> The interval arithmetic provide a correct enclosure of the exact result.
> Since sign is discontinuous , the continuity is definitely lost for future calculation
> exactly in the same way a the result of a interval arithmetic calculation may provide a heavy overestimation of the the exact result.
Again, computed results may be disappointingly pessimistic. Disappointing in the sense that a mathematician knows better than the software.
sign(x) is discontinuous on any interval containing zero, including infinite intervals, but its range is [-1, 1]. Hence, y === 0, no matter what x is, and y is as continuous as you can get.
We are not wrong to flag y as possibly discontinuous, but we aspire to recognize y is continuous. We'd LIKE our software to reflect our knowledge.
If I change to
y = a * f(x)
there is a sort of second-tier discontinuity: for 0 in x, y is discontinuous UNLESS a = [0, 0], so there is a discontinuity in discontinuity at a = 0.
>
> The rule must be
> the system must not lie
> Nothing more.
I agree with "Thou shalt not lie" as a requirement. At the same time, can't we aspire to better?
Forgive me if I play with words. This week, I'm helping to give a dissertation writers boot camp for 25 PhD students. One of the tasks is to help science and engineering PhD students appreciate the power of language as the English and theology majors do. I've just sent an civil engineering student off to spend the morning reading poetry and to report back what insights he gained. We'll see :-)
George
>
> Best regards,
>
> Dominique
>
>> I am becoming a static-ist.
>>
>> George
>>
>> On May 25, 2011, at 6:17 AM, Dan Zuras Intervals wrote:
>>
>>
>>>> Date: Tue, 24 May 2011 19:05:38 -0500
>>>> From: Ralph Baker Kearfott <rbk@xxxxxxxxxxxx>
>>>> To: Nate Hayes <nh@xxxxxxxxxxxxxxxxx>
>>>> CC: John Pryce <j.d.pryce@xxxxxxxxxxxx>,
>>>> stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
>>>> Subject: Re: Ar we succeeding?
>>>>
>>>> All,
>>>>
>>>> Does someone else also have an opinion concerning this (please)?
>>>>
>>>> Baker
>>>>
>>>>
>>> Baker, et al,
>>>
>>> I find I do have an opinion on this. And as long as Nate's
>>> motion is in its discussion period, now is as good a time
>>> as any to discuss it.
>>>
>>> Let me put it in the form of a question I put to Nate:
>>>
>>> Is your philosophy about decorations a tracking approach or
>>> a static approach?
>>>
>>> There seem to be two schools of thought about the meaning
>>> of decorations.
>>>
>>> There is the TRACKING school in which decorations are the
>>> maximal (most pessimistic) result of the tree of evaluations
>>> that led up to the result to which they are attached. That
>>> is, every exceptional or noteworthy incident in that tree is
>>> recorded for all to see whether it is relevant to the final
>>> result or not.
>>>
>>> Then there is the STATIC school in which decorations are
>>> information concerning the current result only. Earlier
>>> decorations may pass through to this result if they still
>>> apply & may be discarded if they do not. In this case the
>>> decoration must be able to be interpreted in the context
>>> of the final result whatever happened before.
>>>
>>> (In either school, the decoration must be ordered WRT to
>>> subsets of arguments. I believe this is both necessary &
>>> sufficient for an FTDIA to be proved.)
>>>
>>> I asked this question of Nate because his motion seemed to
>>> be primarily of the tracking school but with some static
>>> features thrown in.
>>>
>>> I think we need to be consistent on this point. As much
>>> for our own understanding as to explain the meaning of
>>> decorations to the rest of the world.
>>>
>>> I will admit that I started out in the Tracking school.
>>> But some remarks I've heard in this forum & privately have
>>> suggested to me that the Static school might serve us better
>>> as a standard.
>>>
>>> So I ask of all of you: Which philosophy should we espouse?
>>> Tracking or Static?
>>>
>>> I believe that once we decide this many of our more
>>> difficult questions will fall out as obvious.
>>>
>>> Yours,
>>>
>>> Dan
>>>
>>
>> Dr. George F. Corliss
>> Electrical and Computer Engineering
>> Marquette University
>> P.O. Box 1881
>> 1515 W. Wisconsin Ave
>> Milwaukee WI 53201-1881 USA
>> 414-288-6599; GasDay: 288-4400; Fax 288-5579
>> George.Corliss@xxxxxxxxxxxxx
>> www.eng.mu.edu/corlissg
>>
>>
>>
>
>
> --
> Dr Dominique LOHEZ
> ISEN
> 41, Bd Vauban
> F59046 LILLE
> France
>
> Phone : +33 (0)3 20 30 40 71
> Email: Dominique.Lohez@xxxxxxx
>
Dr. George F. Corliss
Electrical and Computer Engineering
Marquette University
P.O. Box 1881
1515 W. Wisconsin Ave
Milwaukee WI 53201-1881 USA
414-288-6599; GasDay: 288-4400; Fax 288-5579
George.Corliss@xxxxxxxxxxxxx
www.eng.mu.edu/corlissg